Finite Group Databasebeta

← #6497351

Subgroups — #6497351

Order 512.

1546 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic, complemented
2#35289C211yesmaximal, characteristic
3#33293C211yesmaximal, characteristic, complemented
4#33290C211yesmaximal, characteristic, complemented
5#33286C211yesmaximal, characteristic, complemented
6#33277C211yesmaximal, characteristic, complemented
7#30229C211yesmaximal, characteristic, complemented
8#30229C211yesmaximal, characteristic, complemented
9#30204C211yesmaximal, characteristic, complemented
10#30203C211yesmaximal, characteristic, complemented
11#27826C214yesmaximal, complemented
12#27740C214yesmaximal, complemented
13#27725C214yesmaximal, complemented
14#27703C214yesmaximal, complemented
15#21863C211yesmaximal, characteristic
16#21833C211yesmaximal, characteristic, complemented
17#21410C211yesmaximal, characteristic
18#21324C211yesmaximal, characteristic, complemented
19#20472C211yesmaximal, characteristic, complemented
20C2×#532C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#2868C2211yescharacteristic
25#2862C2211yescharacteristic
26#2862C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2858C2211yescharacteristic
32#2640—21no—
33#2640—21no—
34#2629—21no—
35#2628—21no—
36#2570—21no—
37#2570C2212yes—
38#2563C2212yes—
39#2563C2212yes—
40#2550—21no—
41#2550C2212yes—
42#2547—21no—
43#2547—21no—
44#2504C2212yes—
45#2504C2212yes—
46#2441—21no—
47#2441C2212yes—
48#2439—21no—
49#2439C2212yes—
50#2436—21no—
51#2436—21no—
52#2331—21no—
53#2330—21no—
54#2311—21no—
55#2311—21no—
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2101C2214yes—
60#2101C2214yes—
61#2101C2214yes—
62#2101C2214yes—
63#2073C2214yes—
64#2071C2214yes—
65#2025C2214yes—
66#2023C2214yes—
67#1993C2214yes—
68#1993C2214yes—
69#1991C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1990C2211yescharacteristic
80#1990C2211yescharacteristic
81#1990C2211yescharacteristic
82#1988C2214yes—
83#1957C2211yescharacteristic
84#1957C2211yescharacteristic
85#1957C2211yescharacteristic
86#1957C2211yescharacteristic
87#1936C2214yes—
88#1936C2214yes—
89#1934C2214yes—
90#1934C2214yes—
91#1895C2214yes—
92#1875C2214yes—
93#1836C2214yes—
94#1835C2214yes—
95#1823C2214yes—
96#1804C2214yes—
97#1771C2212yes—
98#1771C2212yes—
99#1762C2214yes—
100#1753C2212yes—
101#1753C2212yes—
102#1744C2211yescharacteristic
103#1744C2211yescharacteristic
104#1736C2214yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×#41—21no—
108C23×Q8—21no—
109C22×M16C2312yes—
110#532—24no—
111#532—24no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—24no—
117C2×C4⋊Q8—24no—
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2312yes—
123C2×#127C2311yescharacteristic
124C2×#125—21no—
125C2×#125—21no—
126C2×#125C2312yes—
127C2×#124—21no—
128C2×#124—21no—
129C2×#124—21no—
130C2×#124—21no—
131C2×#124—21no—
132C2×#121—21no—
133C2×#121C2312yes—
134C2×C4×D4—21no—
135C2×C4×D4C2312yes—
136C2×#117—21no—
137C2×#117D411yescharacteristic
138C2×#117D411yescharacteristic
139C2×#117C2312yes—
140C22×C4⋊C4—21no—
141C22×C4⋊C4C2312yes—
142C22×C42C2311yescharacteristic
143#431—24no—
144C2×C4⋊C8C2312yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2312yes—
150C4×M16—24no—
151C2×#97C2312yes—
152C2×#97C2312yes—
153C2×#97C2312yes—
154C2×C4×C8C2312yes—
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2312yes—
160#400—22no—
161#399—22no—
162#398—22no—
163#398—22no—
164#397—22no—
165#397—22no—
166#397—22no—
167#397—22no—
168#396—22no—
169#396—22no—
170#395—22no—
171#393—22no—
172#393—22no—
173#393—22no—
174#393—22no—
175#393—22no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—21no—
181#391—21no—
182#391—21no—
183#391—21no—
184#391—22no—
185#391—22no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390C2312yes—
191#390C2312yes—
192#390C2312yes—
193#390C2312yes—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—22no—
203#389—22no—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#386—22no—
213#386—22no—
214#386—22no—
215#386—22no—
216#384—21no—
217#384—21no—
218#384C2312yes—
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#382—21no—
224#382—21no—
225#382—21no—
226#382C2312yes—
227#382C2312yes—
228#382C2312yes—
229#382C2312yes—
230#382C2312yes—
231#382C2312yes—
232#380—21no—
233#380—21no—
234#380—21no—
235#380—22no—
236#380—22no—
237#380—22no—
238#380—22no—
239#380C2312yes—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2311yescharacteristic
247C4×C4⋊C4C2311yescharacteristic
248C4×C22⋊C4D411yescharacteristic
249C4×C22⋊C4D411yescharacteristic
250C4×C22⋊C4C2311yescharacteristic
251C4×C22⋊C4C2311yescharacteristic
252#376D411yescharacteristic
253#376D411yescharacteristic
254C43C2311yescharacteristic
255C43C2311yescharacteristic
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C2×#41—44no—
273C2×#41—44no—
274C2×#41—44no—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8—21no—
278C22×Q8—22no—
279C22×Q8—22no—
280C22×Q8—44no—
281C22×Q8—82no—
282C22×Q8—82no—
283C22×D4—22no—
284C22×D4—21no—
285C22×D4—21no—
286C22×D4C2412yes—
287C23×C4—22no—
288C23×C4—22no—
289C23×C4C2×D411yescharacteristic
290C23×C4C2×D411yescharacteristic
291C23×C4C2411yescharacteristic
292C2×M16—24no—
293C2×M16—24no—
294C2×M16—42no—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—24no—
304C4⋊Q8—24no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—44no—
308C4⋊Q8—44no—
309C4⋊Q8—44no—
310C4⋊Q8—44no—
311C4⋊Q8—82no—
312#127—22no—
313#127—22no—
314#125—44no—
315#125—44no—
316#125—82no—
317#124—44no—
318#124—82no—
319#124—82no—
320#124—82no—
321#124—82no—
322#121—24no—
323#121—24no—
324#121—44no—
325C4×D4—24no—
326C4×D4—24no—
327C4×D4—44no—
328#117—42no—
329#117—42no—
330#117—44no—
331#117—82no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—44no—
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4C2×D411yescharacteristic
381C2×C22⋊C4C2×D411yescharacteristic
382C2×C22⋊C4C2×D411yescharacteristic
383C2×C22⋊C4C2×D411yescharacteristic
384C2×C42—22no—
385C2×C42—22no—
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42#4114yes—
397C2×C42#4114yes—
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C4⋊C8—24no—
405#102—28no—
406#102—44no—
407#102—44no—
408#102—44no—
409C22⋊C8—24no—
410#97—24no—
411#97—24no—
412#97—42no—
413C4×C8—24no—
414C4×C8—22no—
415C4×C8—22no—
416C4×C8—22no—
417C4×C8—22no—
418C4×C8—42no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445C24—22no—
446C24C22×D411yescharacteristic
447#41—816no—
448C2×Q8—44no—
449C2×Q8—42no—
450C2×Q8—44no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—82no—
460C2×Q8—82no—
461C2×Q8—82no—
462C2×Q8—82no—
463C2×Q8—816no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—24no—
467C2×D4—24no—
468C2×D4—44no—
469C2×D4—44no—
470C2×D4—44no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4#12414yes—
497C22×C4#12414yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C2×#4112yes—
512C22×C4C2×#4112yes—
513C22×C4C2511yescharacteristic
514M16—48no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C82−1+414yes—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—41no—
538C4⋊C4—41no—
539C4⋊C4—41no—
540C4⋊C4—41no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C22⋊C4—41no—
564C22⋊C4—41no—
565C22⋊C4—41no—
566C22⋊C4—41no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C22⋊C4—82no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—82no—
592C42—82no—
593C42#12414yes—
594C42#12414yes—
595C42C2×#4112yes—
596C42C2×#4112yes—
597C42C2×#4112yes—
598C42C2×#4112yes—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23C2×#12412yes—
613C23C23×D411yescharacteristic
614Q8—88no—
615Q8—84no—
616Q8—84no—
617Q8—84no—
618Q8—164no—
619D4—44no—
620D4—44no—
621D4—88no—
622D4—84no—
623D4—84no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—28no—
631C2×C4—28no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—88no—
680C2×C4—88no—
681C2×C4C2×#12412yes—
682C2×C4C2×#12412yes—
683C2×C4C2×#12412yes—
684C2×C4C2×#12412yes—
685C2×C4C2×#12412yes—
686C2×C4#57512yes—
687C2×C4#57512yes—
688C2×C4#57612yes—
689C2×C4#57712yes—
690C2×C4C22×#4111yescharacteristic
691C2×C4C22×#4111yescharacteristic
692C2×C4C2×2−1+412yes—
693C2×C4#58512yes—
694C8—24no—
695C8—42no—
696C8—42no—
697C8—42no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—41no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22C22×#12411yescharacteristic
713C22#349911yescharacteristic
714C22#350011yescharacteristic
715C22C2×#57511yescharacteristic
716C22C2×#57511yescharacteristic
717C22C2×#57611yescharacteristic
718C22C2×#57711yescharacteristic
719C4—22no—
720C4—22no—
721C4—22no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—41no—
728C4—41no—
729C4—41no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4#313614yes—
738C4#314014yes—
739C4#346112yes—
740C4#349512yes—
741C2—42no—
742C2—162no—
743C2#3531211yescharacteristic
744C2C2×#313611yescharacteristic
745C2C2×#314011yescharacteristic
746C2#6248111yescharacteristic
747C2#6248311yescharacteristic
748C2#6254311yescharacteristic
749C2#6254611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #35294C2semidirect
3. #33293C2semidirect
4. #33290C2semidirect
5. #33286C2semidirect
6. #33277C2semidirect
7. #30229C2semidirect
8. #30229C2semidirect
9. #30204C2semidirect
10. #30203C2semidirect
11. #27826C2semidirect
12. #27740C2semidirect
13. #27725C2semidirect
14. #27703C2semidirect
16. #21833C2semidirect
18. #21324C2semidirect
19. #20472C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#352942
#352892
#332932
#332902
#332862
#332772
#302292
#302042
#302032
#278262
#277402
#277252
#277032
#218632
#218332
#214102
#213242
#204722